S8 Rectification and Quadrature of the Spherical Ellipse. 



Now as tan ^ = 

 equation, find 



tan^ B 

 Now as tan ^ = - — ~, we may, eliminating ^ from the last 



cos a; sin *j sec^ B 



cos T = ■ - ■ ^- , 



'v/tan4/3 + tan^j; 

 and ij being the common ordinate of the ellipse and hyperbola, 

 2 _ sin^/3cos^A 



~" cos^ a sin^ X + cos^ /3 cos^ X ' 

 we have also sin "je = sin a sin X. 



Making the necessary substitutions deduced from these 

 equations, we obtain 



* 2 . o. rtan2acos2x+ tan^/Ssin^A^ 

 L sec^ a cos-* A + sec^ p sm^* A J 



or tanT = tanA.sina v'l-g^sm^A ^ ^ ^ ^ ^^g^j 



\/l — sin^g sin^A 

 Let t' be the segment of the second tangent between the 

 point of circular section and the major axe, adopting nearly the 

 same notation as in the latter case, we shall have 

 , sin X cos rj sec^ a 



COS "k ^^^ ^—~~. . . 



'/tan^ a + tan^ ^' 

 tan* a cos^ A 



and tan2^ = 



tan^ a cos^ A + tan^ fl sin* A' 



2*' 



. o tan^/Ssin^A 



tan^ a sec* a cos* A + tan* /3 sec* /j sin* A* 



By the help of the last two equations, eliminating the func 

 tions of i' and ij, we find 



tan A tan* /3 cos^ a. 



tanr' 



sing. 7 ^~''f' 

 V l-sin*s 



(77.) 



sin* A 



, . / j\ e-* sm a sin A cos A ,^ , 



hence tan(T— t')= . ^ . „ ^_ —-. (78.) 



^ '/I— e*sin*A\/l-sin*esin*A ^ ^ 



but this last expression is equal to tan 0, see (60.) ; 

 hence t — t' = w, 



a result precisely similar to (44*.). 



